Check to be sure there is no clogged filter or obstructions in the return air duct or output ducts. It's common for a furnace to cycle off due to overheating if it's not getting enough air across it.
Next, I'd swap out the thermostat. You might just take it off the wall and jump the red and white wires together (assuming an RV is the same as a house furnace? - maybe someone can chime in). That sends a constant signal for heat and rules out the thermostat.
Next, you're into the inner-workings of the furnace where a variety of things could be wrong. How long it runs before shutting down may provide some clues.